CNRS researcher at the university of Paris 13.

I am interested in algebraic topology, algebraic geometry, and anything in between.

Email:

ERC

I am currently carrying on a five year project entitled Foundations of motivic real K-theory, financed by an ERC starting grant (2020). The project is situated in the intersection of homotopy theory, algebraic geometry and arithmetic, putting to work the modern apparatus of higher category theory to tackle classical questions about cohomological and motivic invariants of rings and schemes. Our team comprises of three postdocs:

• Tasos Moulinos
• Guglielmo Nocera
• Ran Azouri

and two Ph.D students:

• Victor Saunier
• Jordan Levin

Some of the work in progress includes:

• The real cyclotomic trace map

  With Jay Shah and Thomas Nikolaus, in preparation.

  We identify the geometric fixed points of real topological cyclic homology in terms of normal L-theory, and deduce a real version of the Dundas-Goodwillie-McCarthy theorem in algebraic K-theory. This is part of work building towards the constructing of efficient trace methods for the study of hermitian K-theory.

• The motivic hermitian K-theory spectrum

  With Baptiste Calmès and Denis Nardin, in preparation.

  We construct motivic spectra representing (homotopy invariant) Grothendieck-Witt and L-theory over general base schemes. When the base scheme is regular Noetherian of finite Krull dimension these motivic spectra represent symmetric Grothendieck-Witt and L-theory.

• Hermitian trace formulas for singularity categories

  With Tasos Moulinos and Ran Azouri

  This project involves generalizing work of Toën and Vezzosi on the conductor formula and the trace of singularity categories for a degeneration of a family of smooth and proper schemes, where now the singularity category is endowed with a Poincaré structure, and the conductor formula is expressed in terms of the motivic trace of the generic and special fibres, which take values in their respective Grothendieck-Witt groups.

• Verdier duality and assembly maps

  With Guglielmo Nocera

  In this project we explore a Poincaré categorical approach to the study of the assembly map in L-theory and Grothendieck-Witt theory, pursuing the ideas apprearing in the fourth installement of the nine author project concerning the assembly map of the circle.

• Topological Hochschild homology for stable and Poincaré categories

With Thomas Nikolaus and Victor Saunier

The goal of this project is to develop a framework for the study of THH as a universal trace theory in the setting of stable categories and generalize this perspective to real THH of Poincaré categories. This project is directly related to trace methods and the Dundas-Goodewillie-McCarthy theorem in the setting of both stable and Poincaré categories.

• Chromatic behaviour of hermitian K-theory

With Jordan Levin

The goal of this project is to understand the chromatic behaviour of Grothendieck-Witt and L-theory. In particular, we wish to prove a "white-shift theorem", stating that L-theory is height preserving at the prime 2.

Hermitian K-theory of stable ∞-categories

• Appendix to

  Stable Moduli spaces of hermitian forms (F. Hebestreit, W. Steimle)

  The Grothendieck-Witt space of a Poincaré ∞-category with a suitable weight structure is descibed in terms of a certain group completion involving only Poincaré objects in the heart. This results is what enables one to express classical Grothendieck-Witt theory of rings in terms of that of a suitable Poincaré ∞-category. In the appendix an alternative root to the main application is proposed via a weight theorem on the level of L-theory.

• Hermitian K-theory of stable ∞-categories IV: Poincaré motives

  This is the fourth instalment of a four-paper nine-author project on hermitian K-theory. It contains the theory of Poincaré motives, localising invariants, and applications to multiplicative properties of Grothendieck-Witt and L-theory. We also revisit and generalize the Shansen splitting phenomenon and use it to show that the universal localising replacement of L-theory coincides with L-theory with universal decoration.

  With Baptiste Calmès, Emanuele Dotto, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, Wolfgang Steimle, in preparation.

• Hermitian K-theory of stable ∞-categories III: Grothendieck-Witt groups of rings

  In this third instalment we prove the main results concerning Grothendieck-Witt and L-theory of rings. In particular, we solve the homotopy limit problem for number rings, calculate the various flavours of Grothendieck-Witt groups of the integers, prove finite generation results for Grothendieck-Witt groups of number rings and show that the comparison map from quadratic to symmetric Grothendieck-Witt theory of Noetherian rings of finite global dimension is an equivalence in suffiently high degrees.

  With Baptiste Calmès, Emanuele Dotto, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, Wolfgang Steimle, submitted (pdf, arXiv).

• Hermitian K-theory of stable ∞-categories II: cobordism categories and additivity

  In this second instalment we develop a cobordism inspired approach to hermitian K-theory and use it to prove most of the core results of this project, including additivity and universality of the Grothendieck-Witt spectrum and its relation to L-theory via the Tate square. As a direct corollary we obtain an extension of Karoubi's fundamental theorem to general rings, affirming a conjecture of Karoubi and Giffen.

  With Baptiste Calmès, Emanuele Dotto, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, Wolfgang Steimle, submitted (pdf, arXiv ).

• Hermitian K-theory of stable ∞-categories I: Foundations

  This is the first instalment of a four-part nine-author paper on hermitian K-theory in the setting of stable ∞-categories equipped with a Poincaré structure. We develop the procedure of deriving quadratic functors which allows to produce the classical setting of forms over rings. We also develop in the detail the general case of ring spectra and construct Poincaré structures on parameterized spectra used to recover visible L-theory. We also make a thorough inverstigation of global structural properties of Poincaré ∞-categories, laying the necessary foundations for the subsequent instalments.

  With Baptiste Calmès, Emanuele Dotto, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, Wolfgang Steimle, Selecta Mathematica, 29.1, 2023, p. 1-269 (pdf, arXiv ).

Arithmetic geometry and rational points

• Rational points on elliptic surfaces and squares represented by products of quadratic forms

  We use Swinnerton-Dyer's method to give sufficient conditions for the existence of rational points on K3 surfaces which are 2-coverings of the projective plain ramified over a union of three diagonal conics.

  In preparation.

• Supersolvable descent for rational points

  We develop a technique for deducing results about rational points on quotients of varieties by actions of finite supersolvable groups by reducing the problem to the fibration method over tori.

  With Olivier Wittenberg, Submitted.

• Rational points on fibrations with few non-split fibres

  We revisit the foundations of the fibration method and extract some new unconditional cases when the number of non-split fibres is at most 3.

  With Dasheng Wei and Olivier Wittenberg, Journal für die reine und angewandte Mathematik , 2022.791, 2022, p. 89-133 (pdf, arXiv).

• The Massey vanishing conjecture for number fields

  We prove the n-fold Massey vanishing conjecture for all number fields and all n's.

  with Olivier Wittenberg, Duke mathematical Journal, Advance Publication, 1-41, 2022 (pdf, arXiv).

• Appendix to

  Number fields with prescribed norms (C. Frei, D. Loughran and R. Newton)

  with Olivier Wittenberg, Commentarii Mathematici Helvetici, 97.1,2022,p.133-181 (arXiv).

• Zéro-cycles sur les espaces homogènes et problème de Galois inverse

  with Olivier Wittenberg, Journal of the American Mathematical Society , 33 (3), 2020, p. 775–805 (pdf, arXiv).

• Second descent and rational points on Kummer varieties

  Proceedings of the London Mathematical Society, 118 (3), 2019, p. 606–648. (pdf, arXiv).

• Hasse principle for Kummer varieties

  with Alexei Skorobogatov, Algebra and Number Theory, 10.4, 2016, p. 813-841 (pdf, arXiv).

• Geometry and arithmetic of certain log K3 surface

  Annales de l'Institut Fourier, 67 (5), 2017, p. 2167-2200 (pdf, arXiv).

• Integral points on conic log K3 surfaces

  Journal of the European Mathematical Society, 21 (3), 2019, p. 627–664. (pdf, arXiv).

• On the fibration method for zero-cycles and rational points

  with Olivier Wittenberg, Annals of Mathematics, 183 (1), 2015, p. 229-295 (pdf, arXiv).

• The Hardy-Littlewood conjecture and rational points

  with Alexei Skorobogatov and Olivier Wittenberg, Compositio Mathematica, 150, 2014, p. 2095-2111 (pdf, arXiv).

• Singular curves and the étale Brauer-Manin obstruction for surfaces

  with Alexei Skorobogatov, Annales Scientifiques de l'École Normale Supérieure, 47, 2014, p. 765-778 (pdf, arXiv).

• Homotopy obstructions to rational points

  with Tomer Schlank, In: Alexei Skorobogatov (Ed.), Torsors, Étale Homotopy and Applications to Rational Points, LMS Lecture Notes Series 405, Cambridge University Press, 2013, pp. 280-413 (pdf, arXiv).

Higher category theory

• Deligne's conjecture for unital coherent ∞-operads

  We prove a generalization of Deligne's conjecture for arbitrary unital coherent ∞-operads. With Eduard Balzin, in preparation

• The infinitesimal tangle hypothesis

  We prove an infinitesimal version of the tangle hypothesis, namely, that the cotangent complex of the tangle m-cube monoidal (∞,n)-category is free of rank one, in a suitable sense .

  With Joost Nuiten, in preparation

• Obstruction theory for higher categories

  We develop a Postnikov-based obsrtuction theory for (∞,n)-categories.

  With Joost Nuiten and Matan Prasma, submitted (pdf, arXiv).

• Cartesian Fibrations of (∞,2)-categories

  With Andrea Gagna and Edoardo Lanari, submitted (pdf, arXiv).

• Fibrations and lax limits of (∞,2)-categories

  We develope the notions outer and inner (co)cartesian of (∞,2)-categories and use it to define and study various types of lax and homotopy limits for diagrams taking values in an (∞,2)-category.

  With Andrea Gagna and Edoardo Lanari, submitted (pdf, arXiv).

• Bilimits and bifinal objects

  With Andrea Gagna and Edoardo Lanari, Journal of Pure and Applied Algebra, 226.12, 2022 (pdf, arXiv).

• On the equivalence of all models for (∞,2)-categories

  With Andrea Gagna and Edoardo Lanari, Journal of the London Mathematical Society, 106.3, 2022, p. 1920-1982 (pdf, arXiv).

• Gray tensor products and lax functors of (∞,2)-categories

  We give well-behaved construction of the Gray tensor product in the setting of scaled simpicial sets and show that it is a left Quillen bifunctor, and in particular preserves homotopy colimits in each variable.

  With Andrea Gagna and Edoardo Lanari, Advances in Mathematics, 391, 2021 (pdf, arXiv).

• Ambidexterity and the universality of finite spans

  Proceedings of the London Mathematical society, 121 (5), 2020, p. 1121–1170. (pdf, arXiv).

• Lax limits of model categories

  Theory and Applications of Categories, 35, 2020, p. 959–978. (pdf, arXiv).

• The tangent bundle of a model category

  with Joost Nuiten and Matan Prasma, Theory and Applications of Categories, 34, 2019, p. 1039–1072. (pdf, arXiv).

• Quillen cohomology of (∞,2)-categories

  with Joost Nuiten and Matan Prasma, Higher Structures, 3 (1), 2019, p. 17–66, (pdf, arXiv).

• Tangent categories of algebras over operads

  with Joost Nuiten and Matan Prasma, The Israel Journal of Mathematics, 234, 2019, p. 691–742 (pdf, arXiv).

• The abstract cotangent complex and Quillen cohomology of enriched categories

  with Joost Nuiten and Matan Prasma, Journal of Topology, 11 (3), 2018, p. 752-798 (pdf, arXiv).

• Pro-categories in homotopy theory

  with Ilan Barnea and Geoffroy Horel, Algebraic and Geometric Topology, 17 (1), 2017, p. 567-643 (pdf, arXiv).

• An integral model structure and truncation theory for coherent group actions

  with Matan Prasma, The Israel Journal of Mathematics, 221 (2), 2017, p. 511–561 (pdf, arXiv).

• The Grothendieck construction for model categories

  with Matan Prasma, Advances in Mathematics, 281, 2015, p. 1306-1363 (pdf, arXiv).

• Quasi-unital ∞-categories.

  Algebraic and Geometric Topology, 15 (4), 2015, p. 2303-2381 (pdf, arXiv).

Preprints

• The cobordism hypothesis in dimension 1

  We give a proof of the cobordism hypothesis in dimension 1 using the theory of quasi-unital ∞-categories (pdf, arXiv).

• The section conjecture for graphs and conical curves

  We show that the finite descent obstruction controls the existence of rational points on normal crossing singular curves whose components are all of genus 0, by relating the problem to a fix point property of pro-finite groups acting on pro-finite trees (pdf, arXiv).

• The universal property of topological Hochschild homology, Chromatic Homotopy, K-Theory and Functors, CIRM Luminy, January 2023.

• The infinitesimal tangle hypothesis, GdR topologie algébrique, University of Nantes, October 2022.

• The homotopy theory of quadratic forms, SFB lecture, University of Regensburg, February 2021.

• New perspectives in hermitian K-theory (three talks), New perspectives on K- and L-theory, University of Muenster, September 2020.

• Squares represented by a product of three ternary quadratic forms, and a homogeneous variant of a method of Swinnerton-Dyer, Rational points on irrational varieties, Institut Henri Poincaré, June 2019.

• Supersolvable descent, XXIXèmes Rencontres Arithmétiques de Caen: arithmétique des corps et géométrie arithmétique, Île de Tatihou, May 2019.

• Ambidexterity and the universality of finite spans, Séminaire de topologie algebrique, Université de Paris 13, April 2018.

• Zéro-cycles sur les espaces homogènes et problème de Galois inverse, séminaire théorie de nombres de Bordeaux, March 2018.

• Zéro-cycles sur les espaces homogènes et problème de Galois inverse, séminaire théorie de nombres de Jussieu, March 2018.

• Ambidexterity and the universality of finite spans, séminaire de topologie, université de Lille, January 2018.

• Cohomology of higher categories, réunion annuelle du GdR topologie algébrique et applications, October 2017 (pdf).

• Quillen obstruction theory, séminaire homotopie en géométrie algébrique, October 2017 (pdf).

• Zero-cycles on homogeneous spaces, Rational points 2017, Franken-Akademie Schloss Schney, July 2017.

• Quillen cohomology of enriched categories, séminaire de topologie, géométrie et algèbre, Nantes, June 2017 (pdf).

• Deuxième descente et points rationnels sur les surfaces de Kummer, séminaire arithmétique et géométrie algébrique, université de Strasbourg, January 2017.

• Second 2-descent and rational points on Kummer surfaces, Rational points and algebraic geometry, Luminy September 2016 (pdf).

• Higher additivity, higher monoids and the universal property of finite spans., Homotopy Theory Day, Hebrew University of Jerusalem, July 2016 (pdf).

• Abstract representation theory and the cotangent complex formalism, congrès national de la Société Mathématique de France, Université François Rabelais, Tours, June 2016 (pdf).

• The cotangent complex formalism, séminaire de physique mathématique et de topologie algébrique, université de Angers, May 2016 (pdf).

• Pro-categories in homotopy theory, séminaire de Topologie, Université de Lille, April 2016 (pdf).

• Integral Points on log K3 surfaces, séminaire Variétés Rationnelles, ENS, Paris, April 2016 (pdf).

• The cobordism hypothesis in dimension 1, séminaire de topologie, géométrie et algèbre, université de Nantes, March 2016.

• Integral Points on log K3 surfaces, séminaire d'arithmétique et de géométrie algébrique, université Paris-Sud, March 2016 (pdf).

• Pro-categories in homotopy theory, séminaire de topologie algébrique, université Paris 13, January 2016 (pdf).

• Rational points on fibered varieties, Interactions between arithmetic and homotopy, Royal Imperial College, London, September 2015 (pdf).

• The Hasse principle for generalized Kummer varieties, Göttingen-Hannover number theory workshop, Leibniz university of Hannover, July 2015.

• The descent-fibration method for integral points, Arithmetic geometry, Chow groups and rational points, The Euler Institute, St. Petersburg, June 2015 (pdf).

• The descent-fibration method for integral points, Heilbronn seminar, university of Bristol, Bristol, May 2015 (pdf).

• On the fibration method, séminaire Variétés Rationnelles, Institut Henri Poincaré, Paris, February 2015 (pdf).

• Model fibrations and the Grothendieck correspondence for model categories, Higher Geometric Structures along the Lower Rhine V, Radboud university Nijmegen, June 2014.

• From linear equations in primes to the fibration method, The intercity number theory seminar, university of Groningen, October 2013.

• The section conjecture for graphs and applications for singular curves, London-Paris number theory seminar, Royal Imperial College, London, June 2013.

• The Hasse principle for singular curves with applications for smooth surfaces, séminaire Variétés Rationnelles, Institut Henri Poincaré, Paris, March 2013.

• Quasi-unital ∞-categories and the cobordism hypothesis in dimension 1, algebraic topology seminar, Radboud university Nijmegen, November 2012 (pdf)

• Étale homotopy theory, Workshop on Cohomological Methods in Arithmetic Geometry, institut für Mathematik, Zurich, September 2012 (pdf)

• Étale homotopy and Diophantine equations, Workshop on Arithmetic Geometry and Homotopy Theory, Imperial College, London, May 2012 (pdf)

• Homotopy obstructions to rational points, joint mathematics meeting, AMS Special Session on Rational Points on Varieties, Boston, January 2012 (pdf)

• Homotopy obstructions to integral points, Torsors: theory and applications, International Centre for Mathematical Sciences, Edinburgh, January 2011 (pdf)

• The Étale Homotopy Type and the Local-Global Principle, Rational Points 3, Universität Bayreuth, July 2010 (pdf).